Here's Looking at Euclid

The interesting thing about geometry is that it's real in a couple of ways. There's what we call Euclidean geometry, named after an ancient mathematician/philosopher, that we use in everyday life to measure things. In it, parallel lines never converge and the sum of the angles of a triangle always equals 180 degrees.

But we live on the surface of a sphere, which means that the endless plane we imagine when we construct our Eucledean drawings is really curved. And on a curved surface, parallel lines do intersect and the angles of a triangle add up to more than 180 degrees.

As this quote on Math Blab from English mathematician G. H. Hardy suggests, his brother and sister number wonks have constructed several such non-Euclidean geomtries, each of which is perfectly internally consistent despite their significant differences from one another. The only thing that changes between them are their initial assumptions.

For some reason, contemplation of the different descriptions of the world that can hang together and be internally real proves peaceful this evening. Although I presume that if I were a math student attempting to master that understanding for the purpose of an exam or project my serenity might diminish -- just like the distance between two parallel lines drawn on a sphere.